$m$-noncrossing partitions and $m$-clusters
Bibliographic record
Abstract
Let $W$ be a finite crystallographic reflection group, with root system $\Phi$. Associated to $W$ there is a positive integer, the generalized Catalan number, which counts the clusters in the associated cluster algebra, the noncrossing partitions for $W$, and several other interesting sets. Bijections have been found between the clusters and the noncrossing partitions by Reading and Athanasiadis et al. There is a further generalization of the generalized Catalan number, sometimes called the Fuss-Catalan number for $W$, which we will denote $C_m(W)$. Here $m$ is a positive integer, and $C_1(W)$ is the usual generalized Catalan number. $C_m(W)$ counts the $m$-noncrossing partitions for $W$ and the $m$-clusters for $\Phi$. In this abstract, we will give an explicit description of a bijection between these two sets. The proof depends on a representation-theoretic reinterpretation of the problem, in terms of exceptional sequences of representations of quivers. Soit $W$ un groupe de réflexions fini et cristallographique, avec système de racines $\Phi$. Associé à $W$, il y a un entier positif, le nombre de Catalan généralisé, qui compte les amas dans l'algèbre amassée associée, les partitions non-croisées de $W$, et plusieurs autres ensembles intéressantes. Des bijections entre les amas et les partitions non-croisées ont été données par Reading et Athanasiadis et al. On peut encore généraliser le nombre de Catalan généralisé, obtenant le nombre Fuss-Catalan de $W$, que nous noterons $C_m(W)$. Ici $m$ est un entier positif, et $C_1(W)$ est le nombre Catalan généralisé standard. $C_m(W)$ compte les partitions $m$-non-croisées de $W$ et les $m$-amas de $\Phi$. Dans ce résumé, nous donnerons une bijection explicite entre ces deux ensembles. La démonstration dépend d'une réinterprétation des objets du point de vue des suites exceptionnelles de représentations de carquois.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".